Transient-Time Fractional-Space Trigonometry and Application

  • A. G. Radwan
  • Ahmed S. Elwakil
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7663)


In this work, we use the generalized exponential function in the fractional-order domain to define generalized cosine and sine functions. We then re-visit some important trigonometric identities and generalize them from the narrow integer-order subset to the more general fractional-order domain. It is clearly shown that trigonometric functions and trigonometric identities in the transient-time of a non-integer-order system have significantly different values from their steady-state values. Identities such as sin2(t) + cos2(t) = 1 are shown to be invalid in the transient-time of a fractional-order system. Some generalized hyperbolic functions and identities are also given in this work. Application to the evaluation of the step-response of a non-integer-order system is given.


Fractional-Calculus Generalized Exponential function Fractional order systems Generalized Trigonometric Functions Generalized Hyperbolic Functions 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • A. G. Radwan
    • 1
  • Ahmed S. Elwakil
    • 2
  1. 1.Department of Engineering MathematicsCairo UniversityEgypt
  2. 2.Department of Electrical & Computer EngineeringUniversity of SharjahSharjahUAE

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